Classification problems involve detecting patterns in data and using those patterns to assign a data point to a group of similar data points. If that's too abstract, here are some examples of classification problems: analyzing an email to determine whether it's spam; detecting the language of a piece of text; reading an article and categorizing it as finance, sports, politics, opinion pieces, or crime; and determining whether a review of your product posted on Twitter is positive or negative (this last example is commonly called sentiment analysis).
Classification algorithms are tools that solve classification problems. By definition, they are supervised learning algorithms, as they'll always need a labeled training set to build a model from. There are lots of classification algorithms, each designed with a specific principle in mind or...